derive, using the definitions of speed, frequency and wavelength, the wave equation v = f λ

Progressive Waves 🚂

What is a wave?

A wave is a disturbance that travels through a medium, carrying energy from one place to another without transporting matter. Think of a ripple spreading across a pond after you drop a stone 🌊, or the sound of a drum vibrating through the air 🎵.

Key Definitions 📐

Deriving the Wave Equation 🧮

Let’s imagine a wave traveling along a string. In one second, the wave will have moved a certain distance – that’s the speed $v$. During that same second, the wave will have completed $f$ full cycles. Each cycle covers a distance equal to one wavelength $\lambda$. Therefore:

1. Distance travelled in 1 s = speed × time = $v \times 1$ = $v$.
2. Distance covered by one cycle = wavelength = $\lambda$.
3. Number of cycles in 1 s = frequency = $f$.
4. So, total distance travelled = (distance per cycle) × (number of cycles) = $\lambda \times f$.
5. Equating the two expressions for distance: $v = f \lambda$.

In block form, the wave equation is:

$$v = f \lambda$$

Real‑World Analogy 🎡

Picture a train (the wave) moving along a straight track (the medium). - The speed of the train is $v$. - Each carriage represents one wavelength $\lambda$. - The number of carriages that pass a fixed point every second is the frequency $f$. If the train travels 100 m in one second and has 10 carriages, each carriage is 10 m long: $v = 10 \times 10 = 100$ m s⁻¹. That’s exactly the wave equation in action!

Quick Practice Problems ??

1. A sound wave travels at 340 m s⁻¹ and has a frequency of 170 Hz. What is its wavelength?
2. A water wave has a wavelength of 2 m and a frequency of 0.5 Hz. What is its speed?
3. Explain why a higher frequency wave travels faster if the wavelength stays the same.

Answers: 1) $\lambda = v/f = 340/170 = 2$ m. 2) $v = f\lambda = 0.5 \times 2 = 1$ m s⁻¹. 3) Because $v = f\lambda$, if $\lambda$ is constant, increasing $f$ directly increases $v$.

Take‑away Summary 📌

- Speed $v$ tells us how fast a wave moves. - Frequency $f$ counts how many cycles pass a point each second. - Wavelength $\lambda$ measures the length of one cycle. - The fundamental relationship is $v = f \lambda$, linking all three concepts. Remember the train analogy: speed = (carriages per second) × (length of each carriage). That’s the heart of wave physics!